the pka of benzoic acid is 4.2. show mathematically that this acid is 50% ionized. Use results of part a t oexplain why precipitation is incomplete if the pH of an aquesous solution of benzoate ion is lowered only to ph 7 by adding acid.

Thank you

Oh, dear! You're really pulling out all the chemistry stops today, aren't you? Well, let me see if I can entertain you with some mathematical mildness.

To determine the percent ionization of benzoic acid, we can use the formula:

% Ionization = (10^(pH - pKa) / (1 + 10^(pH - pKa))) * 100

Substituting the given pKa of 4.2 and pH of 7 into the formula, we get:

% Ionization = (10^(7 - 4.2) / (1 + 10^(7 - 4.2))) * 100

Now, let's crunch some numbers and simplify:

% Ionization = (10^2.8 / (1 + 10^2.8)) * 100
% Ionization = (630.957 / (1 + 630.957)) * 100
% Ionization = (630.957 / 631.957) * 100
% Ionization = 0.998 * 100
% Ionization = 99.8%

So, mathematically, benzoic acid is approximately 99.8% ionized at pH 7. Quite a lot, wouldn't you say?

Now, onto the explanation of why precipitation is incomplete when the pH of an aqueous solution of benzoate ion is lowered only to pH 7. Well, you see, at pH 7, the concentration of H+ ions is significantly higher compared to the pKa of benzoic acid, which means the acid will start to reassociate and form non-ionized molecules again. Basically, the acid is playing hard to get and not fully letting go of its ions.

This lack of complete ionization means that the solubility of benzoic acid in water decreases, making it more prone to precipitating out of the solution. But don't worry, it's not completely closed-minded! If you lower the pH further, the concentration of H+ ions will increase even more, which can lead to more successful precipitation. It just likes a little extra persuasion.

So, there you have it! Benzoic acid's 50% ionization and its reluctance to form precipitates at pH 7. I hope I've amused and enlightened you, my chemically curious friend!

To determine the percentage of ionization of benzoic acid, we can use the equation for the ionization of a weak acid:

HA ⇌ H+ + A-

Where HA is benzoic acid, H+ is a hydrogen ion, and A- is the benzoate ion.

The equilibrium constant for this reaction is given by:

Ka = [H+][A-] / [HA]

The pKa of benzoic acid is defined as the negative logarithm (base 10) of Ka. Therefore, we have:

pKa = -log(Ka)

Given that pKa = 4.2, we can calculate Ka using the following equation:

Ka = 10^-pKa

Ka = 10^-4.2 = 6.31 x 10^-5

Since the concentration of the benzoic acid (HA) is equal to the concentration of the benzoate ion (A-) in a 0.1 M solution, let's denote the concentration of each species as [HA] = [A-] = 0.1 M.

Now, let's substitute these values into the Ka equation:

6.31 x 10^-5 = [H+][0.1] / [0.1]

Simplifying the equation:

[H+] = (6.31 x 10^-5) / 0.1

[H+] = 6.31 x 10^-4 M

The concentration of the hydrogen ion is given by [H+]. To calculate the percentage of ionization, we need to compare the concentration of the ionized species (A-) with the initial concentration of the acid (HA).

Percentage of ionization = ([A-] / [HA]) x 100%

Since [A-] = [HA] = 0.1 M:

Percentage of ionization = (0.1 / 0.1) x 100% = 100%

From the calculations, we see that the acid is completely ionized, not 50% as mentioned in the question. Therefore, the result contradicts the statement made in the question.

Regarding part b, if the pH of an aqueous solution of benzoate ion is lowered to pH 7 by adding acid, precipitation is incomplete due to the increased concentration of the benzoic acid (HA) resulting from the partial neutralization of the benzoate ion. As we have just shown, benzoic acid is significantly less ionized at pH 7 compared to the original solution. This decrease in ionization leads to decreased solubility of the benzoic acid in the solution, causing incomplete precipitation.